The next question is how to choose the minimal surface . Emparan’s proposal  consists of the following. Suppose the metric on the boundary of asymptotically-AdS spacetime describes a black hole with a Killing horizon at surface . Presumably, the horizon on the boundary is extended to the bulk. The bulk horizon is characterized by vanishing extrinsic curvature and is a minimal -dimensional surface. Thus, one can choose the bulk horizon to be that minimal surface , the area of which should appear in the holographic formula (303). In this construction the Killing horizon is the only boundary of the minimal surface . This prescription is perfectly eligible if one computes the entanglement entropy of a black hole in infinite spacetime. In  it was applied to the entropy of a black hole residing on the 2-brane in the 4d solution of . In  a similar prescription is used to compute entanglement entropy of the de Sitter horizon.
On the other hand, in certain situations it is interesting to consider a black hole residing inside a cavity, the “black hole in a box”. Then, as we have learned in the two-dimensional case, the entanglement entropy will depend on the size of the box so that, in the limit of large , the entropy will have a thermal contribution proportional to the volume of the box. This contribution is additional to the pure entanglement part, which is due to the presence of the horizon . In order to reproduce this dependence on the size of the box, one should use a different proposal. A relevant proposal was suggested by Solodukhin in .
Let a -dimensional spherically-symmetric static black hole with horizon lie on the regularized boundary (with regularization parameter ) of asymptotically anti-de Sitter spacetime AdSd+1 inside a spherical cavity of radius . Consider a minimal d-surface , whose boundary is the union of and . has saddle points, where the radial AdS coordinate has an extremum. By spherical symmetry the saddle points form a -surface with the geometry of a sphere. Consider the subset of whose boundary is the union of and . According to the prescription of , the quantity this proposal has been verified for and .
It should be noted that as far as the UV divergent part of the entanglement entropy is concerned, the two proposals [83, 139] and  give the same result. This is due to the fact that the UV divergences come from that part of the minimal surface, which approaches the boundary of the AdS spacetime. In both proposals this part of the surfaces and is the same. Thus, the difference is in the finite terms, which are due to global properties of the minimal surface.
From the geometrical point of view, the holographic calculation of the logarithmic term in the entanglement entropy is related to the surface anomalies studied by Graham and Witten  (this point is discussed in ).
Living Rev. Relativity 14, (2011), 8
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